Transitivity of Jordan algebras of linear operators: on two questions by Grünenfelder, Omladič and Radjavi (Q1030485)

Let \({\mathcal A}\) be a Jordan algebra of linear operators on a vector space \(V\) over a field of characteristic \(\neq 2\). In this note, the authors prove that (1) if \({\mathcal A}\) is \(2\)-transitive, then it is dense; (2) if \({\mathcal A}\) is \(n\)-transitive with \(n\geq 1\) and \({\mathcal I}\) is a nonzero Jordan ideal of \({\mathcal A}\), then \({\mathcal I}\) is also \(n\)-transitive. Assertion (1) generalizes a previous result by \textit{L. Grünenfelder}, \textit{M. Omladič} and \textit{H. Radjavi} [``Jordan analogs of the Burnside and Jacobson density theorems'', Pac. J. Math. 161, No. 2, 335--346 (1993; Zbl 0811.46052)], who proved (1) under the aditional assumption that the elements in \({\mathcal A}\) are finite rank operators. Assertions (1) and (2) answer two questions posed in the cited article, in which one can also find a result close to (2), proving the \((n-1)\)-transitivity of \({\mathcal I}\) in the case where \({\mathcal A}\) is \(n\)-transitive and \(n>2\).